In a simplified and well studied version of the problem, the FC makes its decision on \(s_j\)  by looking only at the corresponding reports, i.e. \((r_{1,j}, r_{2,j} \dots r_{n,j})\) . This is a reasonable assumption in some applications, e.g., when the components of \(s^m\)  correspond to the state of the observed system at different time instants, and a decision must be made as soon as the reports regarding the state at time \(j\)  are available. In the absence of Byzantines, the optimum way to combine the local decisions according to the Bayesian approach has been determined in [1]}, [2]}, and is known as Chair-Varshney rule. If local error probabilities are symmetric and equal across the nodes, Chair-Varshney rule corresponds to simple majority-bases decision.
